arXiv subjects

Lasse Fischer

Publications and source records attributed to Lasse Fischer.

14 recordsLinked to original sources

arXiv2026

Anytime-valid testing with e-values and confirmatory adaptive designs

Confirmatory adaptive designs were introduced more than 30 years ago and enable for example sample size re-assessments and the selection of treatments, endpoints as well as subpopulations during the course of a clinical trial. Recently, sequential tests based on e-values for an anytime-valid inference have been developed, promising seemingly similar or even more flexibility and utility. In this note, we compare these two independently developed concepts, shedding light on their formal and methodological connections and differences. Specifically, we show that adaptive design tools like conditional error functions and combination tests are formally equivalent to e-value based, anytime-valid sequential tests. However, in spite of their common fundamental intention to bring flexibility into statistical inference, they have quite different emphases: While hypothesis testing with combination tests and conditional error function usually intent to exhaust type I error rates under the offered flexibility, e-value based testing aims on the additional flexibility with regard to optional continuation, the chosen level and, in recent extensions, in the loss functions to be controlled. We also indicate how recent e-value achievements could enrich clinical trial methodology and adaptive design methodology could inspire and improve e-value based testing.

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arXiv2026

Improving online FDR procedures via online analogs of e-closure and compound e-values

In many scientific applications, hypotheses are generated and tested continuously in a stream. We develop a framework for improving online multiple testing procedures with false discovery rate (FDR) control under arbitrary dependence. Our approach is two-fold: we construct methods via the online e-closure principle, as well as a novel formulation of online compound e-values that is defined through donations. This yields strict power improvements over state-of-the-art e-value and p-value procedures while retaining FDR control. We further derive algorithms that compute the decision at time $t$ in $O(\log t)$ time, and we demonstrate improved empirical performance on synthetic and real data.

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arXiv2025

Choosing the nominal level post-hoc with knockoffs using e-values

The knockoff filter is a powerful tool for controlled variable selection with false discovery rate (FDR) control. In this paper, we leverage e-values to allow the nominal FDR level to be switched post-hoc, after looking at the data and applying the knockoff procedure. This approach addresses a significant limitation of standard knockoffs: while frequently used in high-dimensional regressions, they often lack power in low-dimensional and sparse signal settings. One of the main reasons for this is that the knockoff filter requires a minimum number of selections that depends strictly on the nominal FDR level. By utilizing e-values, we can increase the nominal level in cases where the original procedure makes no discoveries, or decrease it to improve precision when discoveries are abundant. These improvements come without any costs, meaning the results of our post-hoc procedure are always more informative than those of the original knockoff filter. We extend this methodology to recently proposed derandomized knockoff procedures and demonstrate its utility in variable selection problems relevant to drug development using real clinical trial data.

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arXiv2025

Bringing Closure to False Discovery Rate Control: A General Principle for Multiple Testing

We present a novel necessary and sufficient principle for multiple testing methods controlling an expected loss. This principle asserts that every such multiple testing method is a special case of a general closed testing procedure based on e-values. It generalizes the Closure Principle, known to underlie all methods controlling familywise error and tail probabilities of false discovery proportions, to a large class of error rates -- in particular to the false discovery rate (FDR). By writing existing methods as special cases of this procedure, we can achieve uniform improvements, as we demonstrate for the e-Benjamini-Hochberg and the Benjamini-Yekutieli procedures, and the self-consistent method of Su (2018). We also show that methods derived using our novel e-Closure Principle generally control their error rate not just for one rejected set, but simultaneously over many, allowing post hoc flexibility for the researcher. Moreover, because all multiple testing methods for the expected loss error metrics covered by our framework are derived from the same procedure, researchers may even choose the error metric post hoc. Under certain conditions, this flexibility even extends to post hoc choice of the nominal error rate.

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arXiv2025

Bringing closure to FDR control: beating the e-Benjamini-Hochberg procedure

False discovery rate (FDR) has been a key metric for error control in multiple hypothesis testing, and many methods have developed for FDR control across a diverse cross-section of settings and applications. We develop a closure principle for all FDR controlling procedures, i.e., we provide a characterization based on e-values for all admissible FDR controlling procedures. A general version of this closure principle can recover any multiple testing error metric and allows one to choose the error metric post-hoc. We leverage this idea to formulate the closed eBH procedure, a (usually strict) improvement over the eBH procedure for FDR control when provided with e-values. This also yields a closed BY procedure that dominates the Benjamini-Yekutieli (BY) procedure for FDR control with arbitrarily dependent p-values, thus proving that the latter is inadmissibile. We demonstrate the practical performance of our new procedures in simulations.

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arXiv2024

Improving Wald's (approximate) sequential probability ratio test by avoiding overshoot

Wald's sequential probability ratio test (SPRT) is a cornerstone of sequential analysis. Based on desired type-I, II error levels $\alpha, \beta$, it stops when the likelihood ratio crosses certain thresholds, guaranteeing optimality of the expected sample size. However, these thresholds are not closed form and the test is often applied with approximate thresholds $(1-\beta)/\alpha$ and $\beta/(1-\alpha)$ (approximate SPRT). When $\beta > 0$, this neither guarantees error control at $\alpha,\beta$ nor optimality. When $\beta=0$ (power-one SPRT), this method is conservative and not optimal. The looseness in both cases is caused by \emph{overshoot}: the test statistic overshoots the thresholds at the stopping time. Numerically calculating thresholds may be infeasible, and most software packages do not do this. We improve the approximate SPRT by modifying the test statistic to avoid overshoot. Our `sequential boosting' technique uniformly improves power-one SPRTs $(\beta=0)$ for simple nulls and alternatives, or for one-sided nulls and alternatives in exponential families. When $\beta > 0$, our techniques provide guaranteed error control at $\alpha,\beta$, while needing less samples than the approximate SPRT in our simulations. We also provide several nontrivial extensions: confidence sequences, sampling without replacement and conformal martingales.

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arXiv2024

An online generalization of the (e-)Benjamini-Hochberg procedure

In online multiple testing, the hypotheses arrive one by one, and at each time we must immediately reject or accept the current hypothesis solely based on the data and hypotheses observed so far. Many online procedures have been proposed, but none of them are generalizations of the Benjamini-Hochberg (BH) procedure based on p-values, or of the e-BH procedure that uses e-values. In this paper, we consider a relaxed problem setup that allows the current hypothesis to be rejected at any later step. We show that this relaxation allows us to define -- what we justify extensively to be -- the natural and appropriate online extension of the BH and e-BH procedures. We show that the FDR guarantees for BH (resp. e-BH) and online BH (resp. online e-BH) are identical under positive, negative or arbitrary dependence, at fixed and stopping times. Further, the online BH (resp. online e-BH) rule recovers the BH (resp. e-BH) rule as a special case when the number of hypotheses is known to be fixed. Of independent interest, our proof techniques also allow us to prove that numerous existing online procedures, which were known to control the FDR at fixed times, also control the FDR at stopping times.

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arXiv2024

Admissible online closed testing must employ e-values

In contemporary research, data scientists often test an infinite sequence of hypotheses $H_1,H_2,\ldots$ one by one, and are required to make real-time decisions without knowing the future hypotheses or data. In this paper, we consider such an online multiple testing problem with the goal of providing simultaneous lower bounds for the number of true discoveries in data-adaptively chosen rejection sets. Employing the recent online closure principle, we show that for this task it is necessary to use an anytime-valid test for each intersection hypothesis. This connects two distinct branches of the literature: online testing of multiple hypotheses (where the hypotheses appear online), and sequential anytime-valid testing of a single hypothesis (where the data for a fixed hypothesis appears online). Motivated by this result, we construct a new online closed testing procedure and a corresponding short-cut with a true discovery guarantee based on multiplying sequential e-values. This general but simple procedure gives uniform improvements over the state-of-the-art methods but also allows to construct entirely new and powerful procedures.

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arXiv2024

Multiple testing with anytime-valid Monte Carlo p-values

In contemporary problems involving genetic or neuroimaging data, thousands of hypotheses need to be tested. Due to their high power, and finite sample guarantees on type-I error under weak assumptions, Monte Carlo permutation tests are often considered as gold standard for these settings. However, the enormous computational effort required for (thousands of) permutation tests is a major burden. In this paper, we integrate recently constructed anytime-valid permutation p-values into a broad class of multiple testing procedures, including the Benjamini-Hochberg procedure. This allows to fully adapt the number of permutations to the underlying data and thus, for example, to the number of rejections made by the multiple testing procedure. Even though this data-adaptive stopping can induce dependencies between the p-values that violate the usual assumptions of the Benjamini-Hochberg procedure, we prove that our approach controls the false discovery rate under mild assumptions. Furthermore, our method provably decreases the required number of permutations substantially without compromising power. On a real genomics data set, our method reduced the computational time from more than three days to less than four minutes while increasing the number of rejections.

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arXiv2024

Asymptotic Online FWER Control for Dependent Test Statistics

In online multiple testing, an a priori unknown number of hypotheses are tested sequentially, i.e. at each time point a test decision for the current hypothesis has to be made using only the data available so far. Although many powerful test procedures have been developed for online error control in recent years, most of them are designed solely for independent or at most locally dependent test statistics. In this work, we provide a new framework for deriving online multiple test procedures which ensure asymptotical (with respect to the sample size) control of the familywise error rate (FWER), regardless of the dependence structure between test statistics. In this context, we give a few concrete examples of such test procedures and discuss their properties. Furthermore, we conduct a simulation study in which the type I error control of these test procedures is also confirmed for a finite sample size and a gain in power is indicated.

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arXiv2024

Sequential Monte-Carlo testing by betting

In a Monte-Carlo test, the observed dataset is fixed, and several resampled or permuted versions of the dataset are generated in order to test a null hypothesis that the original dataset is exchangeable with the resampled/permuted ones. Sequential Monte-Carlo tests aim to save computational resources by generating these additional datasets sequentially one by one, and potentially stopping early. While earlier tests yield valid inference at a particular prespecified stopping rule, our work develops a new anytime-valid Monte-Carlo test that can be continuously monitored, yielding a p-value or e-value at any stopping time possibly not specified in advance. It generalizes the well-known method by Besag and Clifford, allowing it to stop at any time, but also encompasses new sequential Monte-Carlo tests that tend to stop sooner under the null and alternative without compromising power. The core technical advance is the development of new test martingales (nonnegative martingales with initial value one) for testing exchangeability against a very particular alternative. These test martingales are constructed using new and simple betting strategies that smartly bet on whether a generated test statistic is greater or smaller than the observed one. The betting strategies are guided by the derivation of a simple log-optimal betting strategy, have closed form expressions for the wealth process, provable guarantees on resampling risk, and display excellent power in practice.

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arXiv2023

An exhaustive ADDIS principle for online FWER control

In this paper we consider online multiple testing with familywise error rate (FWER) control, where the probability of committing at least one type I error shall remain under control while testing a possibly infinite sequence of hypotheses over time. Currently, Adaptive-Discard (ADDIS) procedures seem to be the most promising online procedures with FWER control in terms of power. Now, our main contribution is a uniform improvement of the ADDIS principle and thus of all ADDIS procedures. This means, the methods we propose reject as least as much hypotheses as ADDIS procedures and in some cases even more, while maintaining FWER control. In addition, we show that there is no other FWER controlling procedure that enlarges the event of rejecting any hypothesis. Finally, we apply the new principle to derive uniform improvements of the ADDIS-Spending and ADDIS-Graph.

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arXiv2023

ADDIS-Graphs for online error control with application to platform trials

In contemporary research, online error control is often required, where an error criterion, such as familywise error rate (FWER) or false discovery rate (FDR), shall remain under control while testing an a priori unbounded sequence of hypotheses. The existing online literature mainly considered large-scale designs and constructed blackbox-like algorithms for these. However, smaller studies, such as platform trials, require high flexibility and easy interpretability to take study objectives into account and facilitate the communication. Another challenge in platform trials is that due to the shared control arm some of the p-values are dependent and significance levels need to be prespecified before the decisions for all the past treatments are available. We propose ADDIS-Graphs with FWER control that due to their graphical structure perfectly adapt to such settings and provably uniformly improve the state-of-the-art method. We introduce several extensions of these ADDIS-Graphs, including the incorporation of information about the joint distribution of the p-values and a version for FDR control.

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arXiv2022

The Online Closure Principle

The closure principle is fundamental in multiple testing and has been used to derive many efficient procedures with familywise error rate control. However, it is often unsuitable for modern research, which involves flexible multiple testing settings where not all hypotheses are known at the beginning of the evaluation. In this paper, we focus on online multiple testing where a possibly infinite sequence of hypotheses is tested over time. At each step, it must be decided on the current hypothesis without having any information about the hypotheses that have not been tested yet. Our main contribution is a general and stringent mathematical definition of online multiple testing and a new online closure principle which ensures that the resulting closed procedure can be applied in the online setting. We prove that any familywise error rate controlling online procedure can be derived by this online closure principle and provide admissibility results. In addition, we demonstrate how short-cuts of these online closed procedures can be obtained under a suitable consonance property.

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